A Hierarchy of Standard Polynomial Programming Formulations for the Maximum Clique Problem
Sergiy I. Butenko, Mykyta Makovenko, Miltiades Pardalos · SIAM Journal on Optimization · 2022
The maximum clique problem is a classical optimization problem which finds many important applications. Some of the most theoretically interesting and computationally successful approaches to this problem are based on the Motzkin--Straus formulation, expressing the clique number in terms of the maximal value of a standard quadratic program. This intriguing relationship has also motivated significant developments in quadratic optimization, copositive programming, and complexity in nonlinear optimization. Inspired by such developments, this paper establishes a hierarchy of polynomial programming formulations (\textbfP$^k$), $k\in\{2,\ldots, \omega\}$ for the maximum clique problem, relating the clique number $\omega$ of a given graph to the global maximal value of a multilinear polynomial function $f_k(x)$ of degree $k$ over the standard simplex in $\mathbb{R}^{|V|}$. The case of $k=2$ corresponds to the Motzkin--Straus formulation, and the hierarchical feature is in the fact that the set of local maxima of (\textbfP$^{k+1}$) is a subset of the set of local maxima of (\textbfP$^{k}$), $k\in\{2,\ldots, \omega-1\}$. In particular, every local maximum of (\textbfP$^{\omega}$) is global.